Smooth-mask vanishing-moment criterion

Suppose an integrable orthonormal scaling function has a low-pass filter of a multiresolution analysis that is $C^{p-1}$ near $\pi$, and its Fourier transform is $C^{p-1}$ near zero. If the associated wavelet has $p$ integrable vanishing moments, then $m^{(k)}(\pi )=0$ for $k<p$. Indeed, moment differentiation of the Fourier transform makes ${\hat{\psi }}^{(k)}(0)=0$, while $|\hat{\phi }(0)|=1$. In $\hat{\psi }(2t)=e^{-it}\overline{m(t+\pi )}\hat{\phi }(t)$, division by the nonzero smooth factor proves the conclusion. With only continuity of that factor one still obtains a Peano zero, but not automatically higher ordinary derivatives.

Assume the scaling function Fourier transform and the MRA low-pass filter are $C^{p-1}$, and $m^{(k)}(\pi )=0$ for $k<p$. For every nonzero integer $j$, write $j=2^r\ell$ with $\ell$ odd. Iterating the scaling refinement equation at $2\pi j+t$ supplies a factor $m(\pi \ell +t/2^{r+1})$. All its derivatives of order less than $p$ vanish at $t=0$, by periodicity. The product rule gives ${\hat{\phi }}^{(k)}(2\pi j)=0$ for $k<p$.